LECTURE 7

Connection Coefficients

The last of the four basic questions to be considered is the problem of connection coefficients between two sequences of functions. If {p„(x)} and {<?„(x)} are given (n = 0,1, • • •) we wish to find the coefficients ck „ satisfying

 

 

Usually (but not always) p„(x) and qn(x) are polynomials of degree n, in which case there is no question of the existence of In this degree of generality nothing useful can be said about the connection coefficients, and in all instances I know, very little of any interest can be said unless the sets of functions are similar, a notion which will not be made precise. For example, when considering orthogonal polynomials the true intervals of orthogonality should be the same.

 

gives those of the first kind, and

 

 

The required generating functions are

 

The first instance of connection coefficients goes back to Stirling [1]. He defined two sets of numbers, Stirling numbers of the first and second kind:

those of the second kind. There is no agreement on a standard notation, so that in the standard handbook of Abramowitz and Stegun [1] is the notation used above. These numbers are useful in combinatorial problems, but they do not play a role in the problems in these lectures and so will not be mentioned further.

For the classical polynomials there are two very old results which can be derived easily from generating functions:and (7.7)

An interesting generalization of generating functions of the type (7.6) has recently been given by Rota and coworkers (see Mullin-Rota [1] and Rota- Kahaner-Odlyzko [1]). They define polynomials of binomial type as polynomials which satisfy

 

All such polynomials have formal generating functions of the form

 

(7.9)    = /(0) = 0, /'(0)#0.

„ = 0

Examples include p„(x) = x" (hence the name binomial type) and the polynomials £f~1(x) = n\L~l(x), which are defined by the generating function (7.6) when oc = -1.

More general polynomials are the Sheffer sets (s„(x)} relative to a binomial set {p„(x)}.They satisfy

 

and have the generating function

 

(7.11)  I =        g(0)#0,

« = o

and the same f(r) as in (7.9). For two Sheffer sets, Rota-Kahaner-Odlyzko [1] found the connection coefficients by use of umbral calculus (or symbolic methods) which they justify by means of linear operators (see also Brown-Goldberg [1]). There are many interesting formulas in these papers, and these techniques should be learned by anyone who wants to work with polynomial sets defined by simple exponential type generating functions. It is quite likely that the most interesting applications will come from considering polynomials in several variables (see Parrish [1] and the sections in Rota-Kahaner-Odlyzko [1] on cross-sequences and Steffensen sequences).

 

Our main interest is Jacobi polynomials, and they are not included in the above theory. Formula (7.5) is an instance of the explicit construction of connection coefficients between two Jacobi polynomials. For Legendre polynomials this formula was proved by Laplace [1] and his proof extends immediately to ultra­spherical polynomials once the generating function (3.16) is given. One simple application of (7.5) is the following:

 

 

Gegenbauer [1] stated the more general formula

 

 

Initially he stated this only when A - n was an integer, but later [3] he states it for general A and /i. When the limit as n tends to zero is taken and

 

 

is used, then (7.5) is obtained. While (7.5) is easy to prove, it is a little harder to prove (7.13). There are three reasonably natural ways to prove it. The most straightforward is to write C^(x) as a polynomial in xk, and then expand xk in a series of Cj(x). The resulting coefficients can be evaluated by the Chu-Vander- monde sum (see Hua [1, § 7.1] for the details of this argument, but not for the reference to Chu or Vandermonde). As is customary in this field, he derived this formula directly, and he was probably unaware of the long history behind (7.16). Until recently I was also unaware exactly how old (7.16) is. The usual reference is Vandermonde, with occasionally the date 1770 attached. Knuth [1] gave a more precise reference (Vandermonde [1]), but Gasper called my attention to a comment in Chrystal [1]. Chrystal says that (7.16) is usually attributed to Vandermonde, but it is older. How much older he does not say, but I would guess that he thought Euler had it earlier. I have been unable to find it in Euler's work before about 1775. but this is not a very strong proof that he did not have it before 1772. However, Chrystal was right— it had been published by Chu [1] in 1303.1 have not seen this fascinating book, but Needham [1, p. 138] gives the sums

 

In more standard notation these sums are

Putting them together gives

and

 

 

 

It is likely that Chu only had this sum for integer values of p and q, but it is easy for us to conclude the same equality for complex p and q from his result. For both sides are rational functions of p and q which agree infinitely often. Thus Chu really had the value of the general polynomial 2Fi when x = 1. He also had a special case of Saalschutz's formula (see Takacs [1] and Carlitz [1]). Since most mathematical historians have missed these important results in Chu [1], this book should be translated so that mathematicians who cannot read Chinese can see what other treasures are contained in it. The fact that Chu had the "Pascal triangle" property of binomial coefficients is not surprising. It is a fairly obvious fact once the binomial coefficients are discovered. The Chu-Vandermonde sum (7.16) is much deeper, and not at all obvious. The distinction between these two results is really the difference between

 and

 

and then use the Quadratic transformations

 

 

This seems a small difference, but to obtain (7.16) from this sum one must also know how to multiply polynomials of arbitrary degree and collect terms. This is far from obvious until adequate notation has been developed. And the special case of Saalschutz's formula that Chu had was absolutely incredible. To see this one only need look at the contortions some very good mathematicians went through to prove this in the middle of the twentieth century (see the papers in the bibliography of Takacs [1]). Chu did not have the benefit of integral or differential calculus, the tools used by most of these people. He must have been a remarkable mathematician.

 

My favorite proof of (7.13) is first to calculate the coefficients inand

 

 

on the series (7.14) when fi = to derive (7.13). The connection coefficients in (7.19) are very easy to derive by orthogonality and Rodriques' formula (2.1). Explicitly they will be given in (7.33). This method has the disadvantage of having to break a problem into two cases when it should not be necessary to do this, but that is a minor objection.

 

This gives a partial explanation of a fact which has interested me for years. Well- poised series are series

 

When x is set equal to one in (7.19) the resulting formula gives a special case of Dougall's formula. It is

in which numerator and denominator factors can be paired so that their sums are constant. After Rummer's sum of the well-poised at x = — 1 and Dixon's sum of the well-poised 3F2 at x = 1, most of the well-poised series which can be summed are what I like to call "very well-poised", one of the numerator para­meters is one more than the corresponding denominator parameter. This comes in very naturally from the orthogonality relation for Jacobi polynomials. For

 

 

In a Jacobi series this appears in the denominator, so factors of the form (2a)„(a + l)„/(a)„, where a = (a + p + l)/2, tend to occur. This is only a partial explanation. These sums are fundamental results which can be approached from many ways and there is probably an explanation from each of these ways (see Burchnall-Lakin [1] for a partial explanation of this from the point of view of differential equations).

The third method generalizes the second in that one calculates the coefficients in

 

 

This can be done in many different ways, two of them being a use of Rodrigues' formula and the expansion first in terms of (1 — xf and then expanding that in terms of           The resulting coefficients are 3F2's and when <5 = p or y = a

 

Next use Rodrigues' formula

 

and the differentiation formula

 

to obtain

 

or y = ô, (x = p they can be evaluated as products of gamma functions by use of standard formulas. Since this method gives a more general formula than the other two methods we shall give one version of it. The coefficients can be obtained by orthogonality,

There are a number of ways of evaluating the integral in (7.27). The easiest is to use the definition of Jacobi polynomials as hypergeometric functions and integrate term by term. Since there are a number of different representations as hypergeo­metric functions there are various 3F2 representations of ak n. These are all derivable from any one of them by means of the Thomae-Whipple transformation formulas. These transformation formulas are very useful, as we saw in Lecture 6.

 

When y = a this 3F2 reduces to a 2Fi and so is summable by Gauss' formula

 

Using (2.2) in (7.27) leads to

or since it is a polynomial we only need the Chu-Vandermonde sum (7.16). When P = <5 the 3F2 is summable by Pfaff's formula

 

Observe that the coefficients in (7.33) and (7.34) are nonnegative when y > a > — 1, and in (7.32) they are nonnegative when <5 = /? — 1, fi - 2, • ■ • , <5 > -1. The nonnegativity of these coefficients in (7.33) and (7.34) has been a useful fact (see Askey-Wainger [1], Askey-Gasper [5] and Lecture 8), so the problem of deciding when these coefficients are nonnegative for general a, /?, y, <5 is a natural one. Just as in the last lecture the problem reduces to the nonnegativity of a 3F2.

And finally when a = /? and y = <5 the 3F2 is summable by Watson's formula

 

Explicitly the formulas are

 

One powerful method of attacking such problems is to set up recurrence rela­tions. Gauss showed that the general 2Fi(a,b;c;x) can be linearly connected with any two other 2F, 's whose parameters differ from (a, b, c) by integers. He gave the explicit formulas for the contiguous ^'s, i.e., those which differ in only one parameter and by one in this parameter, and the general result follows by iteration. The general 3F2{a, b,c;d,e;x) also has recurrence relations, but this time some of them connect a function and three contiguous functions. See Rainville [1] for methods of obtaining these relations. Four-term recurrence relations are not too easy to handle, so it was only after Watson [3] observed that three-term relations could be found for 3F2's when x = 1 that it was possible to obtain some relativelycomplete answers to questions about the positivity of 3F2's. We shall probably never know exactly what question Watson was trying to solve when he discovered this important fact, but a reading of his paper [3] shows that it was something connected with special Jacobi polynomials. Bailey [4] gave a more systematic treatment, and his methods were adapted in Askey-Gasper [2] to set up two three- term recurrence relations and they have been analyzed to obtain fairly complete results on the nonnegativity of the 3F2's 'n (7.28). Before stating this result another type of problem will be considered, since it leads to the same problem and suggests some possible answers.

 

When X is the unit circle, then

 

Schoenberg [2] proved that the continuous positive definite functions on S\ the k-dimensional unit sphere; x\ + • • • + , = 1, are given by

 

Since the unit circle can be isometrically embedded in S* Schoenberg [2] remarked that

 

Also since S can be isometrically embedded in Sm when k < m he also observed that

 

Given two metric spaces X and Y we wish to see if X can be isometrically embedded in Y. This problem in metric geometry was attacked by Schoenberg in the following way. A function f(r), t 2: 0, is said to be positive definite with resDect to X if

 

 

for all real Cj and points Xj e X. px{x, y) is the distance between x and y measured by the metric on X. If X can be isometrically embedded in 7, then each function which is positive definite with respect to Y is also positive definite with respect to X. When the metric space X is the real line it is an immediate consequence of Bochner's theorem that the bounded, continuous positive definite functions are Fourier cosine transforms of positive measures of finite mass

when 2k and 2n are positive integers. Observe that both of these results are clear from (7.5) and (7.13). Gegenbauer's formula (7.13) had been forgotten by the time Schoenberg wrote [2], so (7.37) was far from clear. Schoenberg's remark was one

of the reasons I suspected a formula like (7.13) was true, and why I noticed it when reading Gegenbauer's paper [3]. It is not surprising that (7.13) was overlooked for so long because Gegenbauer has hundreds of formulas in his papers and very few of them are used. There may be other important formulas there which have been over­looked, but they will probably not be appreciated until they are rediscovered and used. As one further example. Feldheim's formula (3.30) for the special case X = 0 had been given by Gegenbauer [3]. Periodic checks should be made of these old papers to see what has been rediscovered that was once known but was forgotten because no one knew how to us it. One can hope to be lucky and find a formula which is just what is needed, but my experience is that this almost never happens. You find a formula in an old paper only after you have rediscovered it.

Back to Schoenberg's problem of isometric embeddings and positive definite functions. There are other compact manifolds which are quite similar to spheres, the projective spaces over the reals, the complex numbers, the quaternions and a two-dimensional projective space over the Cayley numbers. These are all the compact two-point homogeneous spaces. They have the property that a group of motions exists which takes any pair of points (x,, y,) to (x2, y2) when dist (x,,y,) = dist (x2, y2). Bochner [1] proved that the positive definite functions on these spaces are given by

 

 

where q>„(cos 0) are suitably normalized zonal spherical functions. For the above manifolds these functions are Jacobi polynomials. Cartan [1] proved this for complex projective spaces and Helgason [1] gave the differential equations in general. Gangolli [1] then pointed out the following facts evplicniv.

 

 

In all cases c„ is chosen to give the spherical function the normalization <p„(0) = 1. Here L is the diameter of the space and d is the "real" dimension, that is the dimension as a real manifold. So P16(Cay) has real dimension 16 and dimension 2 over the Cayley numbers. These formulas are nice except for the spherical function of F'(R), the real projective space. These are usually thought of as spheres with antipodal points identified, which is the reason for giving the spherical functions as above. However, it is clearly better to write this line as

 

as can be done by use of the quadratic transformation (3.13).

Since the real numbers can be isometrically embedded in the complex numbers, which can be isometrically embedded in the quaternions, and then on to the Cayley numbers, it is possible to isometrically embed F'(R) in P2d(C), P2d(C) in P4<i(H), and P8(H) in P16(Cay). Using Schoenberg's remark about the reverse inclusion among the positive definite functions we see that

The condition yP — Sol ^ 5 — y + a — /} puts (y, 5) below the line connecting (a, /J) and ( — 1, - 1). It is necessary as is easily seen by considering the case n = 1 in (7.17). The condition y - 6 2a. - 2/? puts (>•, <5) below the line with slope one passing through (2a + 1,2j8 + 1). When <5 ^ /} this condition is also necessary, as can be seen by using an asymptotic formula of Fields [1] and the 3F2 in (7.28) (for further results see Askey-Gasper [2]).

These sums can be inverted (see Askey [1]), and the more general problem

 

 

may also be interesting (see Askey-Gasper [2]).

uu

 

with ak_„ ^ 0 when (a, 0) = (j/2, - j), {J + 1,0) and (3,1), j = 0,1, ■ ■ ■ . Also since P^R) can be isometrically embedded in Pi + '(R), etc., this gives

 

with ak n ^ 0 when y > a; and a,/?,y are suitably restricted. By (7.33) this always holds, so it was natural to conjecture that there was a generalization of (7.38). There is ; it holds for a ^ /? ^ 0, and even stronger results are true. One of those given in Askey-Gasper [2] is as follows. Theorem 7.1. Let

 

The most interesting problem still open of this simple type is to see if P4t(H) can be isometrically embedded in P*k(C). Positive definite functions can be used to show that P2k{C) cannot be isometrically embedded in P2k(R), but they cannot be used to solve the same problem for quaterionic and complex projective spaces.The interest in this problem comes from a desire to have a definite way of telling if one metric space can be embedded in another. When the embedding is in Hilbert space von Neumann and Schoenberg [1] have shown that the positive definite functions are a sufficiently large class of invariants to solve the isometric embedding problem. The above problem might provide a negative answer for an interesting class of metric spaces. On the other hand, if the isometric embedding of P4t(H) in P4k(C) is possible, then this says something interesting about the extension of the complex numbers to the quaternions. Either way the question has an interesting answer.

Before leaving the simple connection problem for orthogonal polynomials two results of a different type should be mentioned. Askey [5] proved a theorem giving positive connection coefficients when certain conditions on recurrence coefficients are satisfied. The proof is similar to the proof of Theorem 5.2.

Wilson [2] proved the following theorem.

 

could be proven for a Si 0. It is very unlikely that (7.41) holds, for the coefficients arise from two applications of (7.40) and one of

 

The coefficients in (7.42) satisfy (-1)" kak „ ;> 0, so there is no hope that the co­efficients in (7.41) are nonnegative.

Theorem 7.2. Let p„(x) be orthogonal with respect to dot(x) and qn(x) be orthogonal with respect to dfi(x). Let the highest coefficients of pn(x) and q„(x) be positive. Then

 

 

with ak n g 0 if

 

 

He has applied this theorem to obtain a few results on an interesting class of discrete orthogonal polynomials which approximate Legendre polynomials better than the Hahn polynomials Q„(x ; 0,0, N) (see Wilson [3]). In Lecture 6 we used

 

 

with c„ > 0, dn > 0 when a,/? > -1 to prove Theorem 6.1. Theorem 6.3 was stronger than Theorem 6.1 so it seems not unlikely that there is a result which is stronger than (7.40). A proof of Theorem 6.3 for at ^ 0 could be given if

with „ ^ 0 when a è 101, P > - l.j = 0,1, • ■ ■ . In general this is still open, but it is true for ft = ±j,a = -j,0, • • • . An equivalent result with an ultraspherical polynomial on the right was proved by Dunkl [1]. His argument is similar to Schoenberg's argument when he proved (7.37), but he used the unitary group rather than the orthogonal group to obtain the Jacobi polynomials on the left side of (7.44). Theorem 6.4 is an immediate consequence of this result (see Askey [12]). Since it would be interesting to prove Theorem 6.4 for a 2: — j, it would be useful if (7.44) could be proved without the restrictions put on by use of group theoretic methods. The next step is clearly to translate this problem into a problem on hypergeometric functions. A straightforward calculation shows that ak „ is a positive multiple of

 

 

From Dunkl's result we know these 4F3 have the right sign behavior when /} = ± j, a = - j, 0, i, ■ • • . For the application to Theorem 6.4 it is sufficient to choose any value of p. The choice a = P is a natural one, for then (7.43) and (7.44) are identical. When a = — 1 the 4F3's become Saalschiitzian, and Whipple's transformation of a Saalschiitzian 4F3 to a well-poised 7F6 can be used to rewrite the sum in many different ways. But I have no faith in the more general conjecture that (7.44) holds when a = — 1. However, this seems to be a very reasonable conjecture when at g - j, and it should be considered in some detail. At present we know very little about a 4F3 which is not well-poised or Saalschiitzian, and this problem might be a wedge which helps us ask the right questions about a more general 4F3.

Surprisingly there is a result of this type when a slight change has been made. Recall that

 

with ak „ 0 when a g |/}|, ß > —I. We can now ask if

 

The first 3F2 which was proven to be nonnegative without either evaluating it or it being a 3F2 with only positive terms was

 

This was treated by Lorentz and Zeller [1]. There may now be enough different special cases so that there is some hope of synthesizing them. However, it is unlikely that the general 3F2 will ever be treated since there are too many parameters. Previous work on the location of zeros of pFq's is summarized in Szego [9, Chap. 6] and Hille [1]. The location of zeros of 0Fq, lF1 and 2Fi is fairly well understood. Only a start has been made on the general lF1(a, b, c; x) but this looks possible (see Steinig [1], Gasper [8], Makai [2], Askey-Steinig [1]). Beyond this there is little hope of solving the general problem, but it looks as if this type of problem will continue to occur, so new methods will have to be developed.

Added in proof. Feldheim [2] contains a number of formulas of the type given in this lecture and a few of a more general type. For example, he gives the connection formula between Pj,y,,)(l — /<(1 - x)) and Pjf-^x). A new proof and an extension of the Lorentz-Zeller inequality (7.47) is given in Askey, Gasper and Ismail [1].